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Tetrads in General Relativity

VI. Physical Geometry

Fock–Ivanenko Coderivatives

   Geometric objects may lose their covariant character under point-dependent Lorentz transformations, in particular, spacetime derivatives of covariant objects. The general procedure to restore proper tensor behavior, is to introduce linear connections. These entities belong to the subgroup ${\text{S}}{{\text{O}}^ + }(1,3)$ and are called Lorentz connections or spin connections.

   Lorentz connections may be characterized as 1-forms acting in the Lorentz algebra:

6.7

\[{{\bs{\omega }}_\mu }(x) : = \frac{1}{2}{\Sigma _{\mu ab}}(x){J^{ab}}\]

The anti-symmetric quantities ${J^{ab}} = - {J^{ba}}$ are the generators of the appropriate representation (6.6) of the Lorentz subgroup ${\text{S}}{{\text{O}}^ + }(1,3)$. The coefficients ${\Sigma _{\mu ab}}\,$, anti-symmetric in the Latin indices, are the spin connection coefficients defined in (4.2). These fields transform inhomogenously according to (4.9). In the present context, they may be looked upon as gauge fields with one spacetime index and two group indices.

   The Lorentz connections (6.7) permit the introduction of the Fock–Ivanenko (FI) coderivative operator, first proposed by Vladimir Fock and Dmitri Ivanenko in 1929:

6.8

\[{\mathcal{D}_\mu } := {\partial _\mu } + {{\bs{\omega }}_\mu } = {\partial _\mu } + \frac{1}{2}{\Sigma _{\mu ab}} {J^{ab}}\]

The importance of the FI coderivative is that it can be defined for all tensorial and spinorial fields. This may be illustrated by the following three examples:

6.9

  1. ${J^{ab}} = {M^{ab}}: \qquad \qquad \quad   {\mathcal{D}_\mu } = {D_\mu } = {\partial _\mu } + {\Sigma {_\mu }{^b}}{_a}$

    With the generator (6.5), the Lorentz connection (6.7) reduces to the spin connection (4.2) as defined in the tensorial formalism of GRT.

  2. ${J^{ab}} = {{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b}: \qquad \qquad {\mathcal{D}_\mu } = {D_\mu } = {\partial _\mu } + {{\bs{\omega }}_\mu }$

    The Lorentz connection (6.7) is then identical to the GA bivector (4.5).

  3. ${J^{ab}} = \frac{1}{2}{\gamma ^a} \wedge {\gamma ^b}: \qquad \quad {\mathcal{D}_\mu } = {\partial _\mu } + \frac{1}{4}{\Sigma _{\mu ab}}{\gamma ^a} \wedge {\gamma ^b}$

    The $\gamma$'s are 4x4 Dirac matrices. This form of the Lorentz connection is relevant to the Dirac equation and -theory in curved spacetime.